let X, X1, X2 be set ; :: thesis: for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <--> f2) <//> f = (f1 <//> f) <--> (f2 <//> f)
let Y, Y1, Y2 be complex-functions-membered set ; :: thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <--> f2) <//> f = (f1 <//> f) <--> (f2 <//> f)
let f be PartFunc of X,Y; :: thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <--> f2) <//> f = (f1 <//> f) <--> (f2 <//> f)
let f1 be PartFunc of X1,Y1; :: thesis: for f2 being PartFunc of X2,Y2 holds (f1 <--> f2) <//> f = (f1 <//> f) <--> (f2 <//> f)
let f2 be PartFunc of X2,Y2; :: thesis: (f1 <--> f2) <//> f = (f1 <//> f) <--> (f2 <//> f)
set f3 = f1 <//> f;
set f4 = f2 <//> f;
set f5 = f1 <--> f2;
A1: dom ((f1 <--> f2) <//> f) = (dom f) /\ (dom (f1 <--> f2)) by Def48;
A2: dom (f1 <--> f2) = (dom f1) /\ (dom f2) by Def46;
A3: dom ((f1 <//> f) <--> (f2 <//> f)) = (dom (f1 <//> f)) /\ (dom (f2 <//> f)) by Def46;
( dom (f1 <//> f) = (dom f1) /\ (dom f) & dom (f2 <//> f) = (dom f2) /\ (dom f) ) by Def48;
hence A4: dom ((f1 <--> f2) <//> f) = dom ((f1 <//> f) <--> (f2 <//> f)) by A1, A3, A2, Lm1; :: according to FUNCT_1:def 11 :: thesis: for b₁ being object holds
( not b₁ in dom ((f1 <--> f2) <//> f) or ((f1 <--> f2) <//> f) . b₁ = ((f1 <//> f) <--> (f2 <//> f)) . b₁ )
let x be object ; :: thesis: ( not x in dom ((f1 <--> f2) <//> f) or ((f1 <--> f2) <//> f) . x = ((f1 <//> f) <--> (f2 <//> f)) . x )
assume A5: x in dom ((f1 <--> f2) <//> f) ; :: thesis: ((f1 <--> f2) <//> f) . x = ((f1 <//> f) <--> (f2 <//> f)) . x
then A6: x in dom (f1 <//> f) by A3, A4, XBOOLE_0:def 4;
A7: x in dom (f1 <--> f2) by A1, A5, XBOOLE_0:def 4;
A8: x in dom (f2 <//> f) by A3, A4, A5, XBOOLE_0:def 4;
thus ((f1 <--> f2) <//> f) . x = ((f1 <--> f2) . x) /" (f . x) by A5, Def48
.= ((f1 . x) - (f2 . x)) /" (f . x) by A7, Def46
.= ((f1 . x) /" (f . x)) - ((f2 . x) /" (f . x)) by RFUNCT_1:14
.= ((f1 <//> f) . x) - ((f2 . x) /" (f . x)) by A6, Def48
.= ((f1 <//> f) . x) - ((f2 <//> f) . x) by A8, Def48
.= ((f1 <//> f) <--> (f2 <//> f)) . x by A4, A5, Def46 ; :: thesis: verum